Bernoulli's Equation¶
Thus far we have considered the motion of a fluid particle along a streamline and derived expressions for the acceleration of the particle along and normal to the flow direction. As previously noted Newton's second law, $F=m\vec{a}$, governs fluid motion. Having already considered acceleration we now turn our attention to the forces acting on the fluid element. These comprise the pressure forces and the gravitational forces and are illustrated below for a fluid particle of volume $\delta { \rlap{V}-} = \delta s\delta n \delta y$ and mass $\delta m$.
Analysis of forces acting on fluid particle. The terms $p$ and ${ \rlap{V}-}$ are the pressure of the particle and its volume. The top row shows the volume of the fluid particle, the areas where the streamwise and normal pressures are exerted and the decomposition of the gravitational forces into streamwise and normal components.The sum of the forces in the streamwise direction is
\begin{equation}
\sum \delta F_s = \delta m~a_s = \delta mV \frac{\partial V}{\partial s} = \rho \delta { \rlap{V}-}~V\frac{\partial V}{\partial s}
\label{motion}\tag{1}
\end{equation}
The gravitational force (particle weight) acting in the streamwise direction:
\begin{equation*}
\delta W_s = -\delta W \sin{\theta} =-\gamma \delta { \rlap{V}-}\sin{\theta}
\end{equation*}
where $\gamma = \rho g$ is the specific weight of the fluid.
The pressure force acting in the streamwise direction:
\begin{equation*}
\delta F_{ps} = (p-\delta p_s)\delta n\delta y - (p+\delta p_s)\delta n\delta y = -2 \delta p_s \delta n\delta y.
\end{equation*}
Since:
\begin{equation*}
\delta p_s \approx \frac{\partial p}{\partial s}\frac{\delta s}{2}~\Rightarrow
\delta F_{ps} = -\frac{\partial p}{\partial s}\delta s\delta n\delta y
=-\frac{\partial p}{\partial s}{\delta \rlap{V}-}
\end{equation*}
Summing the pressure and gravitational forces:
\begin{equation}
\sum \delta F_s = \delta W_s + \delta F_{ps} = \left( -\gamma \sin{\theta}-\frac{\partial p}{\partial s} \right)\delta{ \rlap{V}-}
\label{F_s}\tag{2}
\end{equation}
Equating the RHS of equation \ref{F_s} with the RHS side of equation \ref{motion}, the equation of motion, we get:
\begin{equation*}
-\gamma \sin{\theta}-\frac{\partial p}{\partial s} = \rho V \frac{\partial V}{\partial s}
\end{equation*}
and rearranging:
\begin{equation*}
\frac{\partial p}{\partial s} + \rho V \frac{\partial V}{\partial s} + \gamma \sin{\theta} = 0
\end{equation*}
Using the chain rule, $dp = \frac{\partial p}{\partial s}ds + \frac{\partial p}{\partial n}dn $ and noting that along a streamline $dn = 0$
\begin{equation*}
\frac{d p}{d s} = \frac{\partial p}{\partial s}
\end{equation*}
The velocity derivative can be rewritten, using the power rule, as:
\begin{equation*}
\frac{d}{d s} \left( \frac{V^2}{2} \right) = V\frac{dV}{ds}
\end{equation*}
Additionally $\sin{\theta} = \frac{dz}{ds}$. Combining the above gives Euler's equation:
\begin{equation}
\frac{dp}{ds} + \rho \frac{d}{d s} \left( \frac{V^2}{2} \right) + \gamma \frac{dz}{ds} = 0
\end{equation}
Simplifying:
\begin{equation*}
{dp} + \frac{1}{2}\rho \left( {V^2} \right) + \gamma dz = 0
\end{equation*}
and integrating yields Bernoulli's first equation along a streamline:
\begin{equation}
p + \frac{1}{2}\rho {V^2} + \gamma z = \text{const}
\end{equation}
